Accelerating Parallel Tempering: Quantile Tempering Algorithm (QuanTA)

TL;DR

This paper introduces QuanTA, a novel algorithm that accelerates parallel tempering in high-dimensional, multimodal sampling problems by using a Gaussian transformation, supported by theoretical analysis and empirical validation.

Contribution

QuanTA is a new algorithm that improves the efficiency and scalability of parallel tempering for high-dimensional target distributions.

Findings

QuanTA accelerates mixing in parallel tempering.

Theoretical analysis shows improved efficiency under regularity conditions.

Empirical results demonstrate effectiveness on canonical examples.

Abstract

Using MCMC to sample from a target distribution, $\pi(x)$ on a $d$-dimensional state space can be a difficult and computationally expensive problem. Particularly when the target exhibits multimodality, then the traditional methods can fail to explore the entire state space and this results in a bias sample output. Methods to overcome this issue include the parallel tempering algorithm which utilises an augmented state space approach to help the Markov chain traverse regions of low probability density and reach other modes. This method suffers from the curse of dimensionality which dramatically slows the transfer of mixing information from the auxiliary targets to the target of interest as $d \rightarrow \infty$. This paper introduces a novel prototype algorithm, QuanTA, that uses a Gaussian motivated transformation in an attempt to accelerate the mixing through the temperature schedule of a parallel tempering algorithm. This new algorithm is accompanied by a comprehensive theoretical analysis quantifying the improved efficiency and scalability of the approach; concluding that under weak regularity conditions the new approach gives accelerated mixing through the temperature schedule. Empirical evidence of the effectiveness of this new algorithm is illustrated on canonical examples.